A compact Riemann surface X of genus g > 1 is said to be phyperelliptic if X admits a conformal involution ρ, called a p-hyperelliptic involution, for which X/ρ is an orbifold of genus p. Here we give a new proof of the well known fact that for g > 4p + 1, ρ is unique and central in the group of all automorphisms of X. Moreover we prove that every two p-hyperelliptic involutions commute for 3p ...

An involution π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the enumeration of involutions which avoid a set Ak of subsequences increasing both in number and in length at the same time. Let Ak be the set of all the permutations 12π3 . . . πk of length k. For k = 3 the only subsequence in Ak is 123 and the 1...

We define a statistic, called weight, on involutions and consider two applications in which this statistic arises. Let I (n)denote the set of all involutions on [n](= {1, 2, . . . , n}) and let F(2n)denote the set of all fixed point free involutions on [2n]. For an involution δ, let |δ| denote the number of 2-cycles in δ. Let [n]q = 1+q+· · ·+qn−1 and let (k)q denote the q-binomial coefficient....

We study generating functions for the number of involutions in Sn avoiding (or containing once) 132, and avoiding (or containing once) an arbitrary permutation τ on k letters. In several interesting cases the generating function depends only on k and is expressed via Chebyshev polynomials of the second kind. In particular, we establish that involutions avoiding both 132 and 12 . . . k have the ...

Distinguished involutions in the affine Weyl groups, defined by G. Lusztig, play an essential role in the Kazhdan-Lusztig combinatorics of these groups. A distinguished involution is called canonical if it is the shortest element in its double coset with respect to the finite Weyl group. Each two-sided cell in the affine Weyl group contains precisely one canonical distinguished involution. In t...

In [GM] Guibert and Mansour studied involutions on n letters avoiding (or containing exactly once) 132 and avoiding (or containing exactly once) an arbitrary pattern on k letters. They also established a bijection between 132-avoiding involutions and Dyck word prefixes of same length. Extending this bijection to bilateral words allows to determine more parameters; in particular, we consider the...

In this paper, a classification of free involutions on 3-dimensional homotopy complex projective spaces is given. By the Z2-equivariant Montgomery-Yang correspondence, we obtain all smooth involutions on S with fixed-point set an embedded S.

Let Ak be the set of permutations in the symmetric group Sk with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns Ak. We present a bijection between symmetric Schröder paths of length 2n and involutions of length n + 1 avoiding A4. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of ste...

In this note we propose a novel algorithm for locating in a text T every occurrence of a strings that can be obtained from a given pattern P by successively applying antimorphic involutions on some of its factors. When the factors on which these involutions are applied overlap, a linear time algorithm is obtained. When we apply the involutions to non-overlapping factors we obtain an algorithm r...

We study the orientation preserving involutions of the orientable 3-dimensional handlebody Hg, for any genus g. A complete classification of such involutions is given in terms of their fixed points.